Answer
The speed of each neutron star is $~~8.2\times 10^4~m/s$
Work Step by Step
Let $R_0 = 1.0\times 10^{10}~m$
We can use conservation of energy to find the kinetic energy of the system when the separation distance is $\frac{R_0}{2}$:
$K_2+U_2 = K_1+U_1$
$K_2-\frac{GM^2}{R_0/2} = 0-\frac{GM^2}{R_0}$
$K_2-\frac{2GM^2}{R_0} = -\frac{GM^2}{R_0}$
$K_2 = \frac{GM^2}{R_0}$
Since both neutron stars have the same mass, each star has half of the kinetic energy of the system.
We can find the speed of each neutron star:
$K = \frac{GM^2}{2R_0}$
$\frac{1}{2}Mv^2 = \frac{GM^2}{2R_0}$
$v^2 = \frac{GM}{R_0}$
$v = \sqrt{\frac{GM}{R_0}}$
$v = \sqrt{\frac{(6.67\times 10^{-11}~N~m^2/kg^2)(1.0\times 10^{30}~kg)}{1.0\times 10^{10}~m}}$
$v = 8.2\times 10^4~m/s$
The speed of each neutron start is $~~8.2\times 10^4~m/s$
